where \(\sigma ^\pm _i = \sigma ^x_i \pm i\sigma ^y_i\)  are the raising and lowering spin-\(1/2\)  operators and \(\sigma _i^{x,y,z}\)  denote the Pauli operators acting on spin \(i\) .
Here, we use open boundary conditions and fix the exchange interaction coupling at \(J=1\) . The values \(h_i\)  are, also, drawn independently
from a random uniform distribution \([-W ,W]\)  and the parameter \(\Delta \)  determines the anisotropy
of the model.
This model is known to undergo a phase transition at a critical disorder strength \(W=W_c= 3.5\pm 0.5\) 
from an ergodic phase to an MBL phase which depends on energy density [1]}, [2]}.
In the current study, we focus on the MBL side of the transition, \(W>W_c\) , in which the existence of LIOMs prevents thermalization.
Using the Jordan‐Wigner transformation [3]}, this model can be mapped to a model of spinless fermions and
we are interested in two different cases when \(\Delta =0\)  and \(\Delta =1\)  which corresponds to
the non-interacting Anderson model and an interacting and disordered fermionic model respectively.
